Let \(G\)  be a finite group and let \(p\)  be a prime. Denote by \(G_p\)  a Sylow \(p\) -subgroup of \(G\) . The right multiplication \(R(G)\)  and the automorphism group \(\hbox{\rm Aut}(G)\)  of \(G\)  are permutation groups on \(G\) . It is easy to see that \(R(g)^\alpha =R(g^\alpha )\)  for every \(g\in G\)  and \(\alpha \in \hbox{\rm Aut}(G)\) , implying \(R(G)\)  is normalized by \(\hbox{\rm Aut}(G)\) . Furthermore, \(R(G)\cap \hbox{\rm Aut}(G)=1\) , and hence we have \(R(G)\hbox{\rm Aut}(G)=R(G)\rtimes \hbox{\rm Aut}(G)\) . This group is called the holomorph of the group \(G\) , denoted by \(\hbox{\rm Hol}(G)\) , that is, \(\hbox{\rm Hol}(G)=\langle R(G),\hbox{\rm Aut}(G)\rangle =R(G)\rtimes \hbox{\rm Aut}(G)\)  (see [1]}).
